From Eq. () and Eq. (), one can see that the differential cross section can be formulated in terms of \(\widetilde{xG}\)  and \(\widetilde{xH}\) . Especially, the coefficients of \(\cos (2\Delta \phi )\)  terms in the above expressions can provide us the direct information of linearly polarized WW gluon distributions, \(\widetilde{xH}\) . If one integrate out \(\phi \) , then the coefficients will disappear and the cross section only dependent on the unpolarized ones, \(\widetilde{xG}\) . This indicates that the two gluon distributions can be extracted by measuring the total cross section and the coefficient of \(\cos (2\Delta \phi )\)  respectively. As argued in Refs. [1]}, [2]}, since we are working in coordinate space, \(\Delta \phi =\phi _u-\phi _v\)  with \(\phi _u\)  and \(\phi _v\)  being the azimuthal angles of \({u}\)  and \({v}\) , while \(P_\perp \)  and \(q_\perp \)  are the conjugate momenta of \({u}\)  and \({v}\) , therefore, it is transparent to explore the properties of \(\widetilde{xH}\)  by measuring the asymmetry of \(P_\perp \)  and \(q_\perp \) . Furthermore, attention should be paid is that, for transversely and longitudinally polarized photons, there exists a sign change for the coefficients of \(\cos (2\Delta \phi )\) , as shown in Eq. () and Eq. (). Especially, if \(Q\approx 0\)  (for quasi-photon scattering process), the coefficient will become a negative one.
